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On free actions of discrete quantum groups

Pekka Salmi

Abstract

R. Ellis showed in 1960 that every discrete group acts freely on its Stone-Cech compactification. We extend this result to discrete quantum groups with low duals. The method of proof is different from the earlier proofs in the classical case, using the definition of freeness given by D. A. Ellwood in the setting of noncommutative geometry.

On free actions of discrete quantum groups

Abstract

R. Ellis showed in 1960 that every discrete group acts freely on its Stone-Cech compactification. We extend this result to discrete quantum groups with low duals. The method of proof is different from the earlier proofs in the classical case, using the definition of freeness given by D. A. Ellwood in the setting of noncommutative geometry.
Paper Structure (4 sections, 6 theorems, 48 equations)

This paper contains 4 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. A new look at the Ellis theorem
  4. The Ellis theorem for discrete quantum groups with low duals

Key Result

Lemma 1

Let $X$ be a set and $f:X\to X$ a function with no fixed points. Then there is a partition $X = A_1\sqcup A_2\sqcup A_3$ such that $f(A_i)\cap A_i = \emptyset$ for every $i = 1, 2, 3$.

Theorems & Definitions (11)

  • Lemma 1
  • Lemma 2
  • Theorem 3
  • proof
  • Lemma 4
  • proof
  • Remark 1
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 1 more